The ideal class groups of dihedral extensions over imaginary quadratic fields and the special values of the Artin \(L\)-function
The author studies the relation between the \(p\)-part of the class group of dihedral extensions and the special values of Artin \(L\)-functions at \(0\). More precisely, let \(l\) be and odd prime, let \(L^+/F^+\) be a dihedral extension of degree \(2l\), with the field \(L^+\) being totally real. For \(m\in F^+\setminus (F^+)^2\) consider the field \(L=L^+(\sqrt{-m})\). Let \(\chi\) be an odd irreducible character of \(\text{Gal}(L/F^+)\) and let \(A_L^\chi\) be the \(\chi\)-part of the scalar extended \(p\)-class group \(A_L=\mathbb{Z}_p[\zeta_l] \otimes Cl_L\{p\}\) of \(L.\) Under a certain technical assumption on \(\chi\) the author expresses the \(p\)-adic valuation of \(|A_L^\chi|\) via the \(p\)-adic valuation of the value of the Artin \(L\)-function \(\mathcal{L}(L/F^+,\chi,0).\) The method of the proof is a combination of the Brauer induction theorem with a theorem by \textit{A. Wiles} [Ann. Math. (2) 131, No. 3, 555--565 (1990; Zbl 0719.11082)].
scientific article; zbMATH DE number 5880857